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Function Reference: stblfit

statistics: paramhat = stblfit (x)
statistics: [paramhat, paramci] = stblfit (x)
statistics: [paramhat, paramci] = stblfit (x, alpha)
statistics: [paramhat, paramci] = stblfit (x, alpha, freq)
statistics: [paramhat, paramci] = stblfit (x, alpha, options)
statistics: [paramhat, paramci] = stblfit (x, alpha, freq, options)

Estimate parameters and confidence intervals for the stable distribution.

paramhat = stblfit (x) returns the maximum likelihood estimates of the parameters of the stable distribution, in the Nolan S0 parameterization, given the data in x. paramhat(1) is the tail index alpha, paramhat(2) is the skewness beta, paramhat(3) is the scale gam, and paramhat(4) is the location delta.

[paramhat, paramci] = stblfit (x) returns the 95% confidence intervals for the parameter estimates. The intervals are Wald intervals from the observed Fisher information.

[…] = stblfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default value.

[…] = stblfit (x, alpha, freq) accepts a frequency vector, freq, of the same size as x. freq must contain non-negative integer frequencies for the corresponding elements in x. By default, or if left empty, freq = ones (size (x)).

[paramhat, paramci] = stblfit (x, alpha, options) specifies control parameters for the iterative algorithm used to compute the ML estimates with the fminsearch function. options is a structure with the following fields and their default values:

  • options.Display = "off"
  • options.MaxFunEvals = 400
  • options.MaxIter = 200
  • options.TolX = 1e-6

The stable density has no closed form; it is evaluated by numerical inversion of the characteristic function, which makes fitting considerably slower than for the closed-form distributions. Censoring is not supported.

The estimates are the maximum-likelihood estimates under the mathematically exact density. MATLAB fits an interpolation-based approximation of the stable density, whose maximum-likelihood estimates deviate from the exact ones by about 10^{-2} (and the resulting confidence intervals by up to roughly 20%); stblfit returns the exact (more accurate) estimates.

Further information about the stable distribution can be found at https://en.wikipedia.org/wiki/Stable_distribution

See also: stbllike, stblpdf, stblcdf, stblinv, stblrnd, fitdist, makedist

Source Code: stblfit

Fit a stable distribution to simulated data

 rand ("seed", 42);
 x = stblrnd (1.5, 0.5, 2, 1, 150, 1);
 [paramhat, paramci] = stblfit (x)
paramhat =

   1.5422   0.3029   1.8256   1.2572

paramci =

   1.3078  -0.1665   1.5150   0.7443
   1.7767   0.7723   2.1363   1.7700